tutorial1_sol

tutorial1_sol - MATH1211/10-11(2)/Tu1sol/TNK Department of...

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MATH1211/10-11(2)/Tu1sol/TNK Department of Mathematics, The University of Hong Kong MATH1211 Multivariable Calculus (2010-11 2nd semester) Solution to Tutorial 1 1. From the equation 2 x - 3 y + 5 z = 30, we see that n = (2 , - 3 , 5) is a normal vector to the plane, and (0 , 0 , 6) is a point on the plane. Now a = (3 , 2 , 0) and b = (0 , 5 , 3) are non-parallel vectors and they are not having opposite directions, but both of them are orthogonal to n because a · n = 0 and b · n = 0. Hence a parametric equation for the plane is given by x ( s,t ) = (0 , 0 , 6) + s a + t b , or x ( s,t ) = (0 , 0 , 6) + s (3 , 2 , 0) + t (0 , 5 , 3). (Note that correct answer is not unique.) 2. (a) It is a circle centered at (0 ,a ) with radius a : (b) The surface is formed by rotating the following circle around the z -axis: 3. The two given lines have parametric equations r 1 ( t ) = ( - 3 , 1 , 5) + t (1 , - 2 , 2) and r 2 ( t ) = (4 , 3 , 6) + t ( - 2 , 4 , - 4), respectively. Here we see that the direction vector for
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tutorial1_sol - MATH1211/10-11(2)/Tu1sol/TNK Department of...

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